3.714 \(\int \frac{1}{\left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=59 \[ \frac{x \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{1}{6};2,\frac{1}{2};\frac{7}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a^2 \sqrt{c+d x^6}} \]

[Out]

(x*Sqrt[1 + (d*x^6)/c]*AppellF1[1/6, 2, 1/2, 7/6, -((b*x^6)/a), -((d*x^6)/c)])/(
a^2*Sqrt[c + d*x^6])

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Rubi [A]  time = 0.0867836, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{x \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{1}{6};2,\frac{1}{2};\frac{7}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a^2 \sqrt{c+d x^6}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(x*Sqrt[1 + (d*x^6)/c]*AppellF1[1/6, 2, 1/2, 7/6, -((b*x^6)/a), -((d*x^6)/c)])/(
a^2*Sqrt[c + d*x^6])

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Rubi in Sympy [A]  time = 20.421, size = 49, normalized size = 0.83 \[ \frac{x \sqrt{c + d x^{6}} \operatorname{appellf_{1}}{\left (\frac{1}{6},\frac{1}{2},2,\frac{7}{6},- \frac{d x^{6}}{c},- \frac{b x^{6}}{a} \right )}}{a^{2} c \sqrt{1 + \frac{d x^{6}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

x*sqrt(c + d*x**6)*appellf1(1/6, 1/2, 2, 7/6, -d*x**6/c, -b*x**6/a)/(a**2*c*sqrt
(1 + d*x**6/c))

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Mathematica [B]  time = 0.765885, size = 341, normalized size = 5.78 \[ \frac{x \left (\frac{26 b c d x^6 F_1\left (\frac{7}{6};\frac{1}{2},1;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{3 x^6 \left (2 b c F_1\left (\frac{13}{6};\frac{1}{2},2;\frac{19}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{13}{6};\frac{3}{2},1;\frac{19}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-13 a c F_1\left (\frac{7}{6};\frac{1}{2},1;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}+\frac{49 c (5 b c-6 a d) F_1\left (\frac{1}{6};\frac{1}{2},1;\frac{7}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{3 x^6 \left (2 b c F_1\left (\frac{7}{6};\frac{1}{2},2;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{7}{6};\frac{3}{2},1;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-7 a c F_1\left (\frac{1}{6};\frac{1}{2},1;\frac{7}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}-\frac{7 b \left (c+d x^6\right )}{a}\right )}{42 \left (a+b x^6\right ) \sqrt{c+d x^6} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(x*((-7*b*(c + d*x^6))/a + (49*c*(5*b*c - 6*a*d)*AppellF1[1/6, 1/2, 1, 7/6, -((d
*x^6)/c), -((b*x^6)/a)])/(-7*a*c*AppellF1[1/6, 1/2, 1, 7/6, -((d*x^6)/c), -((b*x
^6)/a)] + 3*x^6*(2*b*c*AppellF1[7/6, 1/2, 2, 13/6, -((d*x^6)/c), -((b*x^6)/a)] +
a*d*AppellF1[7/6, 3/2, 1, 13/6, -((d*x^6)/c), -((b*x^6)/a)])) + (26*b*c*d*x^6*A
ppellF1[7/6, 1/2, 1, 13/6, -((d*x^6)/c), -((b*x^6)/a)])/(-13*a*c*AppellF1[7/6, 1
/2, 1, 13/6, -((d*x^6)/c), -((b*x^6)/a)] + 3*x^6*(2*b*c*AppellF1[13/6, 1/2, 2, 1
9/6, -((d*x^6)/c), -((b*x^6)/a)] + a*d*AppellF1[13/6, 3/2, 1, 19/6, -((d*x^6)/c)
, -((b*x^6)/a)]))))/(42*(-(b*c) + a*d)*(a + b*x^6)*Sqrt[c + d*x^6])

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(1/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="giac")

[Out]

integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)